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Theorem minmar1fval 22673
Description: First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
minmar1fval.a 𝐴 = (𝑁 Mat 𝑅)
minmar1fval.b 𝐵 = (Base‘𝐴)
minmar1fval.q 𝑄 = (𝑁 minMatR1 𝑅)
minmar1fval.o 1 = (1r𝑅)
minmar1fval.z 0 = (0g𝑅)
Assertion
Ref Expression
minmar1fval 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Distinct variable groups:   𝐵,𝑚   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚   𝑅,𝑖,𝑗,𝑘,𝑙,𝑚
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑚,𝑙)   1 (𝑖,𝑗,𝑘,𝑚,𝑙)   0 (𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem minmar1fval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.q . 2 𝑄 = (𝑁 minMatR1 𝑅)
2 oveq12 7457 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3 minmar1fval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3eqtr4di 2798 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
54fveq2d 6924 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6 minmar1fval.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6eqtr4di 2798 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8 simpl 482 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
9 fveq2 6920 . . . . . . . . . . 11 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
10 minmar1fval.o . . . . . . . . . . 11 1 = (1r𝑅)
119, 10eqtr4di 2798 . . . . . . . . . 10 (𝑟 = 𝑅 → (1r𝑟) = 1 )
12 fveq2 6920 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 minmar1fval.z . . . . . . . . . . 11 0 = (0g𝑅)
1412, 13eqtr4di 2798 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1511, 14ifeq12d 4569 . . . . . . . . 9 (𝑟 = 𝑅 → if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)) = if(𝑗 = 𝑙, 1 , 0 ))
1615ifeq1d 4567 . . . . . . . 8 (𝑟 = 𝑅 → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))
1716adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))
188, 8, 17mpoeq123dv 7525 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))
198, 8, 18mpoeq123dv 7525 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)))) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
207, 19mpteq12dv 5257 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
21 df-minmar1 22662 . . . 4 minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))))
226fvexi 6934 . . . . 5 𝐵 ∈ V
2322mptex 7260 . . . 4 (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) ∈ V
2420, 21, 23ovmpoa 7605 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
2521mpondm0 7690 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = ∅)
26 mpt0 6722 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = ∅
2725, 26eqtr4di 2798 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
283fveq2i 6923 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
296, 28eqtri 2768 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
30 matbas0pc 22434 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
3129, 30eqtrid 2792 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3231mpteq1d 5261 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
3327, 32eqtr4d 2783 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
3424, 33pm2.61i 182 . 2 (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
351, 34eqtri 2768 1 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  c0 4352  ifcif 4548  cmpt 5249  cfv 6573  (class class class)co 7448  cmpo 7450  Basecbs 17258  0gc0g 17499  1rcur 20208   Mat cmat 22432   minMatR1 cminmar1 22660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-1cn 11242  ax-addcl 11244
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-nn 12294  df-slot 17229  df-ndx 17241  df-base 17259  df-mat 22433  df-minmar1 22662
This theorem is referenced by:  minmar1val0  22674
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